Derivative of Constant
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The derivative of constant function is zero. That is , if 'c' is a real number, then
$ \frac{\text{d}[c]}{\text{d}x} = 0$
This shows that the slope is zero that means the line is parallel to the Xaxis.
So the equation of the line is y= b( b is any y coordinate.
slope = m =$\frac{y2y1}{x2 x1}$
slope of AB =$\frac{33}{5 1}$
slope of AB =$\frac{0}{4}$
slope of AB =0
As the slope is zero so the equation of the line AB is y = 3( as the line passes through y=3)
slope = m =$\frac{y2y1}{x2 x1}$
slope of AC =$\frac{33}{1 (1)}$
slope of AC =$\frac{0}{2}$
slope of AC =0
As the slope is zero so the equation of the line AC is y = 3( as the line passes through y=3)
Prove that the derivative of a constant function is zero using the limit definition.
Proof : Let f(x) = y = c ( 'c' be any real number)
f(x + $\triangle x $) = c ( as there is no x in the function)
According to the definition of the derivative of the function
$\frac{\text{d}y}{\text{d}x}=\lim_{\triangle x \rightarrow 0}\frac{f(x + \triangle x)  f(x)}{\triangle x}$
Plug in all the values
$\frac{\text{d}y}{\text{d}x}=\lim_{\triangle x \rightarrow 0}\frac{c  c}{\triangle x}$
$\frac{\text{d}y}{\text{d}x}=\lim_{\triangle x \rightarrow 0}\frac{0}{\triangle x}$
$\frac{\text{d}y}{\text{d}x}=\lim_{\triangle x \rightarrow 0}0$
= 0
Examples of derivative of constant
Function : 1) f(x) = 4 2) f(x) = 3 3) s(t) = π 4) y = 0.8 5) y= 3/4

Derivative: 1) f'(x) = 0 2) f'(x) = 0 3) s'(t) =0 (since π is constant) 4) y' = 0 5) y' = 0

12th grade math
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